Re: Proposed elldata database at the PARI workshop

[Jeroen Demeyer <J.Demeyer@UGent.be>]

Bill Allombert wrote:
> Hello PARI-dev,
> 
> Please find a patch in attachment that implement 2 GP functions
> ellsearch and ellidentify. You will need the elldata package available
> here: <http://pari.math.u-bordeaux.fr/~bill/elldata.tar.gz>
> 
> ellsearch() search the database for conductors, isogeny classes or curves:
> 
> ? ellsearch(121)
> %8 = [["121A1", [1, 1, 1, -30, -76]], ["121A2", [1, 1, 1, -305, 7888]], ["121B1", [0, -1, 1, -7, 10]], ["121B2", [0, -1, 1, -887, -10143]], ["121C1", [1, 1, 0,
> -2, -7]], ["121C2", [1, 1, 0, -3632, 82757]], ["121D1", [0, -1, 1, -40, -221]],
> ["121D2", [0, -1, 1, -1250, 31239]], ["121D3", [0, -1, 1, -946260, 354609639]]]
> ? ellsearch("121A")
> %9 = [["121A1", [1, 1, 1, -30, -76]], ["121A2", [1, 1, 1, -305, 7888]]]
> ? ellsearch("121A1")
> %10 = ["121A1", [1, 1, 1, -30, -76]]
> ellidentify return the name, the minimal model and the coordinate
> change:
> ? ellidentify(ellinit([0,0,0,0,-432]))
> %12 = [["27A1", [0, 0, 1, 0, -7]], [2, 0, 0, 4]]

Thanks for coding this up, it works very well.

Since you asked for extensions:
Maybe you could implement ellinit("40A3") and forell() as prososed
earlier?  I like the idea of these functions, especially forell(), which
could be used to search for a curve with certain properties.

If I'm allowed to add one more wishlist item: a function ellgens(E),
returning the generators of the rational elliptic curve E.  This
shouldn't be too hard to implement using ellidentify() and then looking
up the generators of the curbe.

Hypothetical example:
gp> E = ellinit("9744C2");
gp> ellgens(E)
%40 = [[-11, 14], [-10, 18]]